After one month at a new job you decide to save $50. Each month you plan to increase your savings by five dollars.
a.)Write an explicit formula to model the amount you will save each month.
b.)How much will you save in the seventh month?
To solve this problem, we can use a formula to calculate the amount saved each month. The formula for an arithmetic sequence is given by:
\[a_n = a_1 + (n - 1)d\]
where:
- \(a_n\) is the term we want to find (amount saved in a particular month)
- \(a_1\) is the first term (amount saved in the first month)
- \(n\) is the nth term (number of months passed)
- \(d\) is the common difference (increase in savings each month)
In this case, we know that \(a_1 = 50\) (the amount saved in the first month) and \(d = 5\) (the increase in savings each month).
a.) Writing the explicit formula for the amount you will save each month:
Using the arithmetic sequence formula, we can substitute the given values into the equation:
\[a_n = 50 + (n - 1) \cdot 5\]
Simplifying this equation gives us the explicit formula:
\[a_n = 50 + 5n - 5\]
\[a_n = 45 + 5n\]
b.) How much will you save in the seventh month?
To find the amount saved in the seventh month, we need to substitute \(n = 7\) into the formula:
\[a_7 = 45 + 5 \cdot 7\]
\[a_7 = 45 + 35\]
\[a_7 = 80\]
Therefore, you will save $80 in the seventh month.